Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Fourier transform of $g(x)=x\frac{\partial f}{\partial x}$?

I have a problem with the Fourier transform of the function $g(x)=x\frac{\partial f}{\partial x}$. I need the transform to be itself a function of the Fourier transform of $f(x)$ and I don't know how ...
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40 views

Fourier transform of $\mathrm{rec}(x) =\begin{cases} 1 & \text{if }|x| < 0.5,\\ 0.5& \text{if }|x| = 0.5,\\ 0& \text{if }|x| > 0.5 \end{cases}$

$$\mathrm{rec}(x) =\begin{cases} 1 & \text{if }|x| < 0.5,\\ 0.5& \text{if }|x| = 0.5,\\ 0& \text{if }|x| > 0.5 \end{cases}$$ The Fourier transform of this function is ...
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1answer
48 views

What is the fourier transform of this function?

With $$ f(x) = \frac{1}{p} e^{-\pi x^2/p^2} $$ and $p>0$, I got an answer of $\displaystyle e^{-\pi p^2u^2}$. I just wanted to make sure I got the right answer. If I didn't, I will work through ...
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1answer
90 views

Walsh/Hadamard/Fourier Transform

Hey guys can anyone explain to me what the Walsh/Hadamard/Fourier Transform actually does and how and when do I use it? Can you also recommend me some textbooks that I can use to help me understand it ...
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31 views

$f \in S \rightarrow \hat f \in S$

Let $S$ be the Schwartz space and $ \hat f$ be the Fourier transform of $f$. I hope to prove that $f \in S \rightarrow \hat f \in S$. I know some properties about Fourier transfrom but I do not know ...
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94 views

Polynomial in $D$ is surjective; is the image of a differential closed in $L^2$?

Let $p(x)$ be a polynomial with complex coefficients and let $f$ be a smooth function from $\mathbb{R}$ to $\mathbb{C}$. Let $D$ be the differential operator. Then we can consider the linear map $p(D) ...
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1answer
477 views

Inverse fourier transform 3 dimensions

Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$ As a hint I've been given: Its the unique solution to the equation ...
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102 views

Fouries series of $\sin{\sum_n a_n \sin{n\theta}}$

What's the Fourier series for $\sin({\sum_n a_n \sin{n\theta}})$?
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276 views

A Mathematical way to represent a image kernel?

How to represent the calculation in this image mathematically? For example: With the discrete convolution and Fourier Transform. It tries to do a calculation on the original image (image A/input) ...
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804 views

Fourier analysis prerequisites and lecture notes

I want to know what the prerequisites are for fully grasping Fourier analysis, and some free pdfs and such to help me with it (no videos, actual paper I can print and read/ make exercises at school). ...
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76 views

Fourier coeficients, convergence of the integral

Let $g\in L_{\infty}(\mathbb{T})$. For any $-\pi<a<b<\pi$ let $\chi_{[a,b]}$ be the characristic function of $[a,b]$. Prove that ...
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117 views

Use the full Fourier Transform in x to solve

Use the full Fourier Transform in x to solve: $$ u_{xx} + u_{yy} =0 \, \,\,\, -\infty<x<\infty\,,\,\, 0<y<1$$ $$u(x,0) = \left\{ \begin{array}{l l} 0 & \quad \text{if ...
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1answer
37 views

Terminology of functions in $L_2$

I am reading a text that states ... any function in $L_2(0,\pi)$ has a Fourier sine series that converges to it in $L_2(0,\pi)$ ... Unfortunately no definition of $L_2$ is given. What does ...
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146 views

Identity involving partial sums of Fourier series

Suppose $f$ is a continuous periodic function and $S_Nf(x) = \sum^N_{n=−N} \hat f(n) e^{inx}$, where $$\hat f(n)= \frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-inx} dx.$$ How can I show that ...
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289 views

Decay of Fourier Coefficients and Smoothness

I'm trying to prove that if $|\widehat{f}(k)| = |a_k| < \frac{M}{|k|^{2 + \varepsilon}}$ then $f$ is continuously differentiable. I'm not quite sure how to do this. There was another question on ...
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58 views

Counterexample to Fourier coefficient decay at the bound

There is a proposition in Fourier analysis that states that, for some $\epsilon > 0$, $M > 0$, if $c_k \leq M/|k|^{n+1+\epsilon}$, where $c_k$ is the k-th Fourier coefficient of $f$, then $f \in ...
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36 views

Given $\widehat{\varphi} = \chi_{[-1/2,1/2]}$, show the system $\{\varphi(2^v x)\}_{v \in \mathbb{Z}}$ is not orthogonal

If $$\widehat{\varphi} = \chi_{[-\frac{1}{2},\frac{1}{2}]}$$ then it is not hard to compute via the inverse Fourier transform that $$\varphi(x) = \frac{\sin(\pi x)}{\pi x}$$ so we need to show ...
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330 views

A theorem about Lipschitz regularity and Fourier transform

How to prove that: A function $f$ is uniformly Lipschitz $\alpha$ over $\mathbb R$ if $$\int_{-\infty}^{+\infty}|\hat f(\omega)|(1+|\omega|^\alpha)d\omega<+\infty$$ A function $f$ is uniformly ...
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427 views

Fourier transform $|x|$

Can anyone explain to me why $\mathrm{Fourier}(|x|)=-\sqrt{\frac{2}{\pi}}/w^2$, for $|x|\leq1$? Doing the integral as per definition, I found it to be like this: ...
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131 views

$\int_{-\infty}^{\infty}i\cdot \sin(x)\sin(2{\pi}kx)\;dx$ during Fourier transform

I am trying to do a time-to-frequency domain transform using Fourier transform. My function is very simple: $$ f(x) = \sin(x) $$ By definition its Fourier transform should be: $$ F(k) = ...
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355 views

What is a spherical Gaussian kernel?

In this paper (page 8, Example 3), a spherical Gaussian kernel is defined by the formula $$K(\mathrm x,\mathrm y)=e^{-2\epsilon(1- \mathrm x\cdot\mathrm y)}$$ where $\mathrm{x,y}\in ...
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Discrete time fourier transform of partial sum

I came across the following property of the DTFT: $ \mathcal{F} \Bigg(\sum_{m=- \infty}^{n}x[m]\Bigg) = \frac{1}{1- e^{-j \omega}} X(e^{-j \omega}) + \pi X(e^{-j0}) \sum_{m= ...
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DFT of some functions

Given the DFT pair $x[k]$ and $X[r]$, for a sequence of length N, express the DFT of the following sequences as a function of X[r]: $$ y[k]=x[2k]$$ I guess this is a simple question, but I can't ...
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441 views

Inverse fourier transform of $ 1/(1+s^2)$

Hoi, I want to have the inverse fourier transform $\mathcal{F}^{-1}(\frac{1}{1+s^2})$. So I thought about using some properties of fourier-transform. But knowing the answer I must make some sort of ...
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125 views

Fourier Transform question using a certain property

If $$f(x) = \frac{1}{p}e^{\frac{-\pi x^2}{p^2}},$$ for some $p > 0$, find $F(f(x))$. I'm thinking I need to use the scaling property.
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105 views

Fejer Kernel problem

I encountered this in my notes: $$\int^{\frac{1}{2}}_{-\frac{1}{2}}\mathcal{F}(f- f')df = 1\;\;\;\;\;\;\forall\;f'\;\;\;(1)$$ where $$\mathcal{F}(f) = \frac{\sin^2(N\pi f)}{N\sin^2(\pi f)}.$$ I know ...
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698 views

Intuitively, why is the Gaussian the Fourier transform of itself?

It's a standard exercise to find the Fourier transform of the Gaussian $e^{-x^2}$ and show that it is equal to itself. Although it is computationally straightforward, this has always somewhat ...
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9k views

Prove of the Parseval's theorem for Discrete Fourier Transform (DFT)

If $x[k]$ and $X[r] $ are the pair of discrete time Fourier sequences, where $x[k]$ is the discrete time sequence and $X[r]$ is its corresponding DFT. Prove that the energy of the aperiodic sequence ...
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408 views

Sifting Property of Convolution

This is going to be a dumb question, but I can't figure it out, so here goes $ f(t)\quad \bigotimes \quad \delta \quad (t\quad -\quad { t }_{ o }) $ = $\int { f(\tau )\delta (t\quad -\quad { t }_{ ...
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Help with using IFFT to calculate radial distribution function g(r)

I am trying to use ifft function to evaluate the radial distribution function g(r), r is distance (nm), by using the structure factor s(q) , which is function of wave vector q (1/nm) : $$ g(r) = ...
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number of zeros of complex waves

Does anybody know about any type of methods how to calucalte/estimate the number of the zeros of complex waves (periodic functions as superposition of many harmonic waves) within a given period [0,x] ...
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Prove Parseval for the Fourier transform

Can you please show me how to prove $$\int_{-\infty}^\infty f(x)^2 dx = \frac{1}{2 \pi} \int_{-\infty}^\infty [Ff(x)]^2 dx$$ where $Ff(t) = \displaystyle\int_{-\infty}^\infty f(x) e^{-itx}dx$.
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Solve a differential equation using Fourier series

Assume I have a second order differential equation $\ddot{x} = F(x,\dot{x})$ (or an equivalent equation of first order) and that I know there is a periodic solution to it (for simplicity's sake, ...
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239 views

Fourier Series on a 2-Torus

Taking into account the answer given to this question, in special, the relation between the eigenfunctions of the Laplace-Beltrami operator and the Characters of a group does this imply that on a ...
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Intuition behind Fourier coefficients

Actually I'm trying to dive into Fourier series and have some trouble understanding the idea behind the Fourier coefficients. Let's have a Fourier series $$f(x) = a_0 + ...
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169 views

Fourier Series Expansion of the Partial Differential Equation

Partial differential equation: ...
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195 views

Fourier transform of $\log(f(x))$

Suppose for a function $f(x)$ becomes $F(k)$ after a Fourier transform, what is the Fourier transform of $\log(f(x))$? I cannot find any related formula in Fourier transform table or list properties. ...
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50 views

Fourier Transform in 3-dimension

I am trying to solve this integral in 3-dimensions $$ \int_{\bf{k}} \frac{({\bf q}/2-{\bf k})e^{i{\bf k}\cdot{\bf r}}e^{(-k/\Lambda)^4}}{({\bf q}/2-{\bf k})^2+m^2}.$$ Any suggestion??? Thanks
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Characteristic function for positive part of random variables

I need your help in solving the following problem: I have to calculate the characteristic function for the positive side of a random variable - simplest case: let $Y$ be standard normal and $Y^+ =\max ...
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497 views

Fourier transform of $\sin(2\pi ft)$

I have a function $\sin(2\pi\cdot f\cdot t)$ where $t$ is the time domain and $f$ is the frequency.I must represent the fourier transform of this function in polar and cylindrical coordinates. I can ...
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88 views

Approximation using a Fourier transform with low pass filter

I need to approximate a function f, but I cannot do so with frequencies that exceed 1kHz What is the best approximation I can get? Is taking the Fourier transform then zeroing any term above 1kHz the ...
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What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
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Recursive Inverse Fast Fourier Transform (FFT)

Given a polynomial $A$ in point-value form consisting of the 4 points $(1,5)$, $(i, -1-2i)$, $(-1, -7)$ and $(-i,3-2i)$. Using the recursive inverse FFT algorithm to interpolate, I want to find the ...
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Another aspect of Heisenberg uncertainty principle

In fourier transformation theory, we have the Heisenberg uncertainty principle, i.e. Suppose $\phi$ is a function in [Schwarz space][1] which satisfies the normalizing condition ...
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Fourier transform of Laplace equation with boundary conditions

The function $u(x,y)$ satisfies the partial differential equation $$\nabla^{2}u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0\text{ in }0<y<a, ...
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180 views

Fourier transform with respect to x with partial y derivative

For a function $f(x,y)$ that decays rapidly as $x \rightarrow_{-\infty}^{\infty}$ we define its Fourier transform with respect to $x$ by $\int_{-\infty}^{\infty}f(x,y)e^{-ikx}dx$ a) Prove that if ...
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330 views

how do you do this integral from fourier transform.

I am trying to find the fourier transform of $$\frac{\sin(ax)}{x}$$ for $a >0$. This is clearly an even function so we only need to do the real part, but I could not evaluate ...
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567 views

Fourier transform using the convolution theorem

The function $f(t)$ satisfies the integral equation $f(t)+2\int_{-\infty}^{\infty}H(s)e^{-s}f(t-s)ds=H(t)e^{-t}$ and decays as t $\rightarrow_{-\infty}^{\infty}$ By taking the Fourier transform of ...
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46 views

Retrieving coefficients of polynomial from dft

I am trying to multiply two polynomials using DFT and I don't know how to get the last bit from the DFT of their multiplication. So there's p(x) = x - 4, dft -3, i-4, -5, -i-4 And q(x) = x^2-1, dft ...
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Proof that the Fourier transform of a positive, real, symmetric function is positive, real, symmetric

Let $f(x)$ be a real function, symmetric around $x=0$. From the properties of the Fourier transform, we know that $\hat{f}(\omega)$ (the spectrum of $f(x)$) is also a real function, symmetric around ...